Play B♭ Ionian, then C Dorian and then E♭ Lydian. Do they sound the same because they are comprised of the same notes? Of course not. Why? Because the root in B♭ Ionian, becomes a m7 in Dorian instead of Ionian's M7, and the P4th in Ionian becomes m3 in Dorian instead of Ionian's M3, completely changing the sound and tonality of Dorian vs Ionian: Ionian is the definitive major tonality. Dorian, in modern music, is the definitive minor tonality, because although both contain a P4 and P5, their respective 3rd's and 7ths are different - major vs minor. Same with any mode of any scale, to a greater or lesser extent, depending on the mode relative to the parent scale: Shifting modes == Shifting tonality:
The notes of the different modes of any scale are the same, but because they are played in a different sequence ("mode" = the manner of playing a scale), their sound - their tonality changes. They sound different and they imply different chords entirely. These differences become very significant as you play different modes in different harmonic contexts.
To hear this really well, play C Ionian and then B Locrian, the 7th mode of C Major. You're playing the same notes in both cases- the notes of the C Major scale - but they sound entirely different. Carry this further and think about the diatonic 7th chords derived from each mode: When you play a C Ionian -Major scale, the 7th chord is a CM7. When you play B Locrian scale, the 7th chord is a B half diminished chord ( Bm7b5 ) a very different chord with a tritone, the diminished 5th, at its core. Yet that's all with the same 7 notes of C Ionian - just starting the sequence at a different note. Now you are starting to understand the concept of modes and their use:
Harmony is created through manipulating the relationships of various notes and chords to one another in particular ways, to create different sounds and mixtures of sounds that have different effects on the listener. When you change the order of the notes in question, their harmonic relationships to one another - and the other notes surrounding them - changes - for better, or for worse.
Very simple: If you play a B♭ major scale over an E♭Maj♯11 chord, your root B♭ will be a P 5th in relation to Eb - not necessarily the chord tone you want in that context, etc. (Not the best example. @jdjazz in their more articulate, precise answer gives better examples of how you can run afoul in improvisation if you don't use modes.)
But when you play E♭ Lydian over an E♭Maj♯11 chord, you are playing a scale (mode and scale are in this sense interchangeable) that has the same root as the chord being played, and is a harmonic refection of that chord, to the extent that the chord is derived from that mode. The notes that comprise the chord correspond directly to the notes, at their same intervalic positions, as the mode being played - that's the key to diatonic harmony.
When you play Lydian, the notes that determine the unique tonality of that mode - root/#4/M7 - are exactly the chord tones you are playing, and the other notes of that mode are harmonically related to the chord in way that generally sounds good and "make sense". Another way of saying it is that E♭Maj♯11 is diatonic to E♭ Lydian.
(The late, great jazz/fusion guitarist Allan Holdsworth once said that his great breakthrough came at age of 12, when he realized that chords were derived from scales [and modes - again, interchangeable here] and not vice versa.)
It is true that in many cases, you can get away with just playing the parent scale, but you will be treading on shaky ground - hit or miss - and your playing will not sound well integrated and organic - it will sound boring and not quite right - you'll likely come off as an amateur to "those who know". (Been there, done that...) If you listen to the great jazz players, you'll hear the difference. That's one of the things that tells you someone really knows how to play and isn't just faking it through - they're playing through different modes and scales to reflect the harmonic context of the music.
Listen carefully to what someone like Coltrane or Freddy Hubbard or Horace Silver is doing as they play through the changes of a piece - how they are constantly shifting the harmonies and tonalities as the music moves and develops - they are working modes and scales and chords in many different ways to create a rich and interesting harmonic blend. They seem to do it automatically - until you try to do it yourself. Sure they all have great talent and great ears - but they also worked their butts off - every one them - to figure out how things worked so they could play the way they do. Some of them used the names you'll find in a music textbook, others had their own private language - but they all understood how things hang together.
Like everything in music, your ear must be the ultimate judge, but the rules and recommendations that have been handed down in books or by word of mouth among musicians make sense and they work.